Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.1 → 5.1

Time: 13.4s

Precision: 64

Math

\[x \cdot \left(1 + y \cdot y\right)\]

↓

\[\mathsf{fma}\left(y, y, 1\right) \cdot x\]

C

double f(double x, double y) {
        double r15829617 = x;
        double r15829618 = 1.0;
        double r15829619 = y;
        double r15829620 = r15829619 * r15829619;
        double r15829621 = r15829618 + r15829620;
        double r15829622 = r15829617 * r15829621;
        return r15829622;
}

↓

double f(double x, double y) {
        double r15829623 = y;
        double r15829624 = 1.0;
        double r15829625 = fma(r15829623, r15829623, r15829624);
        double r15829626 = x;
        double r15829627 = r15829625 * r15829626;
        return r15829627;
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.1
Target	0.1
Herbie	5.1

\[x + \left(x \cdot y\right) \cdot y\]

Derivation

1. Initial program 5.1

   \[x \cdot \left(1 + y \cdot y\right)\]

2. Simplified 5.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 1\right) \cdot x}\]

3. Final simplification 5.1

   \[\leadsto \mathsf{fma}\left(y, y, 1\right) \cdot x\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1.0 (* y y))))